Glad to be corrected re: the composition of the vestibules! I've not only
tried to read up on the topic, but also had chronic labyrinthitis and had a doctor try to explain what was happening there — and in neither case was the explanation anywhere near enlightening as the one you just gave :)
> You can only feel, via these organs, the 2nd derivative of position (0th derivative - position, 1st derivative - velocity, 2nd derivative - acceleration)
Well, sure, but the thing that I meant by "builds a picture" isn't just what you're feeling (which, indeed, is acceleration, not position), but also the information the brain has available to predict with.
IIRC, I've read about "hardwired neural logic" (i.e. a brain function outside of our conscious awareness that we have from birth) for flow-state obstacle avoidance in known environments, which involves the brain doing internal dead reckoning based on integrating over its acceleration-picture of the body — thus allowing a person to e.g. make their way home through a forest they've walked through a million times, with their eyes closed, yet stepping over tree roots at just the right time.
Dead reckoning over an integration of the acceleration-picture also seems to come into play with proprioception. Thus why people who are given a local nerve block in an arm, might end up smacking themselves in the face with said arm when trying to e.g. adjust their glasses. They're missing the feedback signals† from the arm that would allow them to compute the acceleration of the arm, and therefore the moment-to-moment position of the arm.
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† I'd love to know how the brain figures out what the acceleration forces being applied to the extremities are. I'm guessing you'd probably be almost the right person to know the answer to that question! (The perfect person would be one of those "prosthetic arm you can feel" researchers.)
I know the brain is getting signals back from the muscles in the extremity — how flexed they are, how much work they're having to do to keep in position — and that's probably a majority of the information needed... presuming you are flexing the muscle.
But we still know where our limbs are in space even when we have them "loose." So there's likely an additional component — the one that allows people without numbness in their extremities, to avoid being smacked in the face by their own hand when someone picks their arm up and drops it above their face while they're asleep.
My naive guess for what that data source could be: pressure on blood vessels. As you move the extremity, the blood drags behind the limb, and so some blood vessels feel an temporary increase in pressure, and others a temporary decrease in pressure. The brain then takes the derivative of the blood pressure data; builds a map of blood pressure deltas; and then "recognizes", in each extremity's picture, the accelerations upon the extremity.